MATH 561: Set Theory I
Fall 2001, MWF 8:00-8:50, 113 McAllister, Schedule 826671
Stephen G. Simpson, 333 McAllister, 863-0775, t20@psu.edu
This introductory set theory course will be suitable for all
mathematics graduate students, and even advanced undergraduates. The
only prerequisite is mathematical maturity.
The textbook will be Set Theory, by Kenneth Kunen (paperback).
Supplementary textbooks will be Set Theory, by Thomas Jech, and
Classical Descriptive Set Theory, by Alexander Kechris.
Set theory has two aspects.
On the one hand, set theory is an interesting branch of mathematics in
its own right, and set-theoretic methods have turned out to be
extremely useful for attacking a variety of problems in several other
branches of mathematics, including analysis, probability, topology,
and combinatorics.
On the other hand, set theory is of great importance as the common
foundation or basis for ALL of mathematics. It is widely accepted
that ALL mathematical concepts and structures, in EVERY branch of
mathematics, are to be defined and explained in terms of sets,
relations on sets, and mappings between sets. Furthermore, a
mathematical proof is commonly regarded as correct if and only if it
can be formalized within axiomatic set theory.
From this foundational or axiomatic point of view, it is particularly
exciting to consider how the standard Zermelo/Fraenkel axioms may be
supplemented or modified by consideration of additional or alternative
set-theoretic axioms. Among these are the Axiom of Choice, the
Continuum Hypothesis, the Generalized Continuum Hypothesis, G"odel's
Axiom of Constructibility, Martin's Axiom, the Axiom of Determinacy,
and the various large cardinal axioms.
MATH 561 will provide an introduction to both aspects of set theory,
the mathematical and the foundational, with emphasis on the
foundational.